$ -3.\overline{8} \div 0.\overline{38} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 10x &= -38.8889...\\ x &= -3.8889...\end{align*} $ $\begin{align*} 9x &= -35 \\ x &= -\dfrac{35}{9}\end{align*} $ $\begin{align*} 100y &= 38.3838...\\ y &= 0.3838...\end{align*} $ $\begin{align*} 99y &= 38 \\ y &= \dfrac{38}{99}\end{align*} $ So, the problem becomes: $ -\dfrac{35}{9} \div \dfrac{38}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{35}{9} \times \dfrac{99}{38} = {?} $ $ \phantom{-\dfrac{35}{9} \times \dfrac{38}{99}} = \dfrac{-35 \times 99}{9 \times 38} $ $ \phantom{-\dfrac{35}{9} \times \dfrac{38}{99}} = \dfrac{-35 \times \cancel{99}11} {\cancel{9} \times 38} $ $ \phantom{-\dfrac{35}{9} \times \dfrac{38}{99}} = -\dfrac{385}{38} $